J. Chem. Theory Comput. 2006, 2, 229-235 229

Detailed Balance in Ehrenfest Mixed Quantum-Classical Dynamics

Priya V. Parandekar and John C. Tully*

Department of Chemistry, Yale UniVersity, P.O. Box 208107, New HaVen, Connecticut 06520

Received August 25, 2005

Abstract: We examine the equilibrium limits of self-consistent field (Ehrenfest) mixed quantum- classical dynamics. We derive an analytical expression for the equilibrium mean energy of a multistate quantum oscillator coupled to a classical bath. We show that, at long times, for an ergodic system, the mean energy of the quantum subsystem always exceeds the temperature of the classical bath that drives it. Furthermore, the energy becomes larger as the number of states increases and diverges as the number of quantum levels approaches infinity. We verify these results by simulations.

1. Introduction total energy is conserved, and both methods have proved Mixed quantum-classical dynamics (MQCD), in which quite accurate in many applications. selected quantum mechanical degrees of freedom are coupled We showed in a previous paper that, for a two-level to a system of classical mechanical degrees of freedom, has quantum system coupled to a many-particle classical bath, proved to be a useful complement to standard classical the “fewest-switches” version of surface hopping13 correctly molecular dynamics (MD) simulations. Quantum effects obeys detailed balancing; the two-level system approaches including electronic transitions,1,2 proton tunneling, and zero- a quantum temperature equal to the classical temperature of point motion3-6 can be introduced within a computationally the bath.21 This is not necessarily the case for the Ehrenfest tractable classical MD framework. A critical requirement for method, as has been discussed by several authors,21-24 the success of a MQCD theory is the proper treatment of signaling a potentially serious deficiency of the method. We the “quantum backreaction”, the altering of the classical previously derived a closed-form expression for the mean forces due to transitions in the quantum subsystem.7-12 Two energy of a two-level quantum system coupled by Ehrenfest widely used approaches for approximating the quantum dynamics to a classical bath, showing that the quantum backreaction have emerged, “surface hopping”13-15 and subsystem approaches a temperature that is finite but higher “Ehrenfest”.16-20 In both approaches, quantum transitions than the temperature of the classical bath to which it is arise in the same way, governed by the time-dependent coupled.21 In this paper, we generalize our previous result Schro¨dinger equation in which the time variation of the to a quantum subsystem composed of an arbitrary number Hamiltonian arises from the motions of the classical particles. of quantum states. For the special case of equally spaced The methods differ only in the way the classical paths evolve. quantum levels, we are able to obtain an exact, closed-form In surface hopping, the forces derive from a single quantum expression for the mean energy of the quantum subsystem. state, subject to sudden stochastic “hops” to different This expression shows, remarkably, that in the limit of an quantum states. The Ehrenfest method is a self-consistent infinite number of equally spaced levels, that is, a harmonic field method; the forces governing the classical particles arise oscillator, the mean energy of the quantum subsystem from a weighted average of quantum states. Both the surface- approaches infinity no matter how low the classical temper- hopping and Ehrenfest methods allow for energy transfer ature that drives it. The populations of each state are not between the quantum and classical subsystems such that the equal in this limit, so the quantum subsystem does not approach infinite temperature. However, the populations * Corresponding author phone: (203) 432-3934; fax: (203) 432- decrease with increasing quantum number sufficiently slowly 6144; e-mail: [email protected]. so that the mean energy diverges. 10.1021/ct050213k CCC: $33.50 © 2006 American Chemical Society Published on Web 01/10/2006 230 J. Chem. Theory Comput., Vol. 2, No. 2, 2006 Parandekar and Tully 2. Two-Level Quantum Subsystem coefficients. Substituting eq 5 into the time-dependent In a previous publication,21 we derived a closed-form Schro¨dinger equation gives a set of coupled differential expression for the equilibrium mean energy of a two-level equations for the time-varying amplitudes cR(t), câ(t), and quantum subsystem coupled to an infinite number of classical cγ(t): particles via the Ehrenfest self-consistent field approximation. i We recast the amplitudes cR and c of quantum levels R and ˘ )- - 3‚ + 3‚ â cR pRcR q dRâcâ q dγRcγ (6a) â into two new variables i ) | |2 ) - | |2 c˘ )-  c + q3‚d c - q3‚d c (6b) X câ 1 cR (1a) â p â â Râ R âγ γ and ˘ )- i - 3‚ + 3‚ cγ pγcγ q dγRcR q dâγcâ (6c) ) * + * Y cRcâ câcR (1b) where we have assumed, for simplicity of notation, that the The variables X and Y can be shown to behave as effective adiabatic wave functions Rq, âq, and γq are real-valued. The classical variables. 25 We derived a classical Liouville nonadiabatic coupling vector dRâ is given by equation for the probability distribution f(q,p,X,Y)ofthe ) 〈R |∇ 〉 positions q and momenta p of the classical particles and the dRâ q qâq (7) variables X and Y. We then obtained the steady-state solution of the Liouville equation, which produced the following Couplings between the other states are defined similarly. simple expression for the equilibrium mean energy of the Note that, because we have chosen the adiabatic representa- quantum subsystem in terms of the temperature, T,ofthe tion, there is no potential energy term coupling the quantum classical bath: levels. Since the Ehrenfest method is invariant to choice of representation,26 the results we derive here apply for any valid - â exp[ â/kBT] representation. From eq 6, we obtain the time derivatives of h)〈| |2〉 ) - E câ â kBT - - (2) 1 exp[ â/kBT] the populations. d where the energy of the lower quantum level R is taken to |c |2 )-q3‚d (c* c + c*c ) + q3‚d (c* c + c*c ) (8a) dt R Râ R â â R γR R γ γ R be zero, â is the energy of the upper level â, and kB is Boltzmann’s constant. It can be shown with some algebra d |c |2 ) q3‚d (c* c + c*c ) - q3‚d (c*c + c*c ) (8b) that the mean energy of the quantum subsystem produced dt â Râ R â â R âγ â γ γ â by Ehrenfest dynamics, as given by eq 2, is always greater than the desired Boltzmann energy d |c |2 ) q3‚d (c*c + c*c ) - q3‚d (c* c + c*c ) (8c) dt γ âγ â γ γ â γR R γ γ R  exp[- /k T] h ) â â B EBOLTZ + - (3) Summing all parts of eq 8 demonstrates conservation of 1 exp[ â/kBT] norm. We define four independent effective classical phase 3. Three-Level Quantum Subsystem space variables for the quantum subsystem, U, S, W, and X, We first generalize this result to a three-level quantum in terms of the quantum amplitudes: subsystem and, later, to an N-level system. We follow the ) | |2 same procedure as above. The three-level quantum subsystem U câ (9) is described in an adiabatic basis. The wave functions Rq, S ) |c |2 (10) âq, and γq are the eigenfunctions of the quantum Hamiltonian γ for fixed classical positions q W ) c*c + c*c (11) R ) R â γ γ â Hq q R(q) q (4a) ) * + * ) X câcR cRcâ (12) Hqâq â(q) âq (4b) The variables U, S, W, and X are the independent variables γ )  (q) γ (4c) Hq q γ q required to characterize a three-state system. Whereas the The subscript q indicates that the quantum Hamiltonian and three complex-valued amplitudes introduce six variables, two | |2 its eigenfunctions depend parametrically on the classical are not independent. The quantity cR is determined by conservation of norm, and the variable positions q. The eigenenergies R(q), â(q), and γ(q), in general, are also functions of q; they are the adiabatic Y ) c*c + c* c (13) potential energy surfaces for states R, â, and γ, respectively. γ R R γ We express the wave function of the quantum subsystem at can be expressed as follows in terms of the independent R any time t as a linear combination of q, âq, and γq variables: ) R + + ψ(t) cR(t) q câ(t) âq cγ(t) γq (5) (x4US - W2x4U - 4U2 - 4US - X2 + WX Y ) (14) where cR(t), câ(t), and cγ(t) are the complex-valued expansion 2U Ehrenfest Mixed Quantum-Classical Dynamics J. Chem. Theory Comput., Vol. 2, No. 2, 2006 231

From eqs 8-12, we obtain equations for the time derivatives Nc - - 2 of the four effective classical variables U, S, W, and X: f(q,p,U,S,W,X) ) A e V(q)/kBT ∏ e pi /2mkBT g(U,S,W,X) (21) i)1 ˙)-3‚ + 3‚ U q dâγW q dRâX (15) where A is a normalization constant and ˙)3‚ - 3‚ S q dâγW q dγRY (16) 1 - g(U,S,W,X) ) (4US - W2) 1/2(4U - 4U2 - ˙) - 3‚ - 3‚ + 3‚ ( 2 W 2(U S)q dâγ q dγRX q dRâY π ( -  ) U( -  ) γ â x - 2 - 2 -1/2 - â R p 4US W (17) 4US X ) exp[ ] kBT S( -  )  X˙)2(1 - 2U - S)q3‚d + q3‚d W - q3‚d Y ( γ R R Râ γR âγ exp[- ] exp(- ) kBT kBT (R -  ) â x4U - 4U2 - 4US - X2 (18) p 1 - ) (4US - W2) 1/2(4U - 4U2 - 2 The two separate branches (() in eqs 17 and 18 can be π - 2 -1/2 - | |2 treated separately and, thereby, pose no mathematical dif- 4US X ) exp[ ( ∑ ci i)/kBT] (22) ficulty. As the quantum subsystem evolves according to the i)R,â,γ time-dependent Schro¨dinger equation, the classical subsystem evolves self-consistently according to Hamilton’s equations Equation 22 is the probability distribution function which of motion. As for the two-level case, the phase space determines any average properties of the three-state quantum variables of the classical subsystem are (q, p) where q and subsystem. Because we have carried out the derivation in p are the positions and momenta, respectively, of the Nc the adiabatic representation in which the Hamiltonian is number of classical particles. For simplicity of notation, we diagonal, the energy of the quantum subsystem depends only | |2 | |2 | |2 assume that the classical variables have been transformed on the probabilities cR , câ , and cγ . We can then into a frame in which the quantum system is coupled to only integrate eq 22 over dW and dX to obtain the un-normalized 27 a single component of momentum, p1. As shown elsewhere, probability distribution in variables U and S using mass-weighted coordinates, this can be achieved for any pair of quantum levels. We assume further that all of ) ∫+x4US ∫+x4U-4U2-4US gU,S(U,S) )-x )-x - 2- g(U,S,W,X)dW dX the nonadiabatic couplings are in the same direction. The W 4US X 4U 4U 4US - - latter is not true in general, but since detailed balance is a U(â R) S(γ R) statement about the forward and backward transition rates ) exp[- ] exp[- ] k T k T between a pair of quantum states, this simplification cannot B B affect the final results. As a result of these simplifying R 1 - ) - | |2 assumptions, the dot products can be replaced by scalar exp( ) exp( ∑ ci i) (23) k T k T )R multiplications in eqs 15-18. The backreaction of the B B i ,â,γ quantum subsystem on the classical subsystem is incorpo- rated as the Hellmann-Feynman force, which acts only on The probability distribution function gU,S(U,S) is a function | |2 classical momentum 1. of only two independent variables, U and S, that is, câ 2 2 and |cγ| , since the population of the ground state |cR| can ∂ ˘ )- V(q) - ∂ 〈 | | 〉 be expressed in terms of the other two populations in p1 ψ(t) Hq ψ(t) ∂q1 ∂q1 accordance with conservation of norm. The simple product form of eq 23 appears deceptively simple. We note that the ∂ )- V(q) - - - - - state populations are not independent as a simple product (γ â)dâγW (â R)dRâX ∂q1 might suggest but are correlated because of the constraints - 2 (R γ)dγRY (19) that 0 < |ci| < 1 for each state i, and the sum over all states 2 of |ci| is unity. Now equipped with the time derivatives of the phase space variables (quantum and classical), we proceed to derive the probability distribution function f(q,p,U,S,W,X), which obeys 4. N-Level Quantum Subsystem the Liouville equation (see McQuarrie28) It is straightforward to generalize eq 23 to obtain the un-

Nc normalized probability distribution for an arbitrary number ∂f pi ∂f ∂V ∂f ∂(fU˙ ) ∂(fS˙ ) ∂(fW˙ ) ) ∑[ - ] + + + + of quantum levels, N: ∂t i)1 m ∂qi ∂qi ∂pi ∂U ∂S ∂W ∂(fX˙ ) ∂f 1 N + - - - - - | |2 | |2 | |2 ) - | |2 [ (γ â)dâγW (â R)dRâX g2,3,..N( c2 , c3 ,... cN ) exp( ∑ ci i) (24) ) ∂X ∂p1 kBTi 1 - ) (R γ)dγRY] 0 (20) 2 with the constraints 0 < |ci| < 1 for each state i, and the 2 The function f(q,p,U,S,W,X) that satisfies eq 20 is sum over all states of |ci| is unity. As for the three-level 232 J. Chem. Theory Comput., Vol. 2, No. 2, 2006 Parandekar and Tully case, the probability distribution functions are correlated required to achieve ergodicity, that is, to ensure that the because of the constraint of conservation of norm. system achieves a true equilibrium independent of the We now derive the mean energy of an N-state quantum classical and quantum initial conditions. This was verified subsystem, using eq 24 for the probability distribution of numerically. The quantum subsystem is an N-level system the populations: coupled to atom 1 of the classical chain. The assumption that the quantum subsystem is coupled only to the first atom 1 -| |2 -| |2-| |2 of the chain is for convenience only; it does not affect any h) ∫ | |2 ∫1 c2 | |2 ∫1 c2 c3 | |2 E { d c2 d c3 d c4 ... 0 0 0 of the conclusions. The quantum energy levels and non- adiabatic couplings are taken to be independent of q. The N-1 N - N - | |2 2 2 1 2 number of classical atoms in the chain was typically chosen ∫1 ∑ ci | | | | | | d cN (∑i ci ) exp( ∑i ci )} i)2 to be 20. A Langevin friction constant γ and white random 0 i)2 kBT i)2 / force F(t) were imposed on atom number Nc of the chain, 1 -| |2 -| |2-| |2 the one most distant from the quantum subsystem, to ensure ∫ | |2 ∫1 c2 | |2 ∫1 c2 c3 | |2 { d c2 d c3 d c4 ... 0 0 0 that the classical subsystem maintained the correct canonical ensemble equilibrium. F(t) is a Gaussian random variable N-1 - N of width given by29 - | |2 2 1 2 ∫1 ∑ ci | | | | d cN exp( ∑i ci )} (25) i)2 -1 1/2 0 kBT i)2 ) σ (2γmkBTδ ) (29)

The ground-state energy 1 is assumed to be zero for where δ is the time step of the integration. The parameters convenience. The integrations limits in eq 25 result from in eqs 28 and 29 were chosen to be the same as those in a conservation of norm; that is, the constraint that the sum of 21 -1 previous publication, V0 ) 175 kJ/mol, a ) 4.0 Å , γ ) all probabilities be unity. Equation 25 is the basic result of 1014 s-1, and m ) 12 amu. The classical equations of motion this paper. The equilibrium mean energy of the quantum were integrated using a modified Beeman algorithm.29,30 subsystem by the Ehrenfest method depends only on the The quantum equations of motion, describing the time energies of the quantum levels and is independent of the evolution of the complex expansion coefficients of the wave coupling strength. It is also clearly not a Boltzmann function ψ distribution, as shown below. In the special case that the energy levels of the quantum N p ˘ ) - p 4‚ subsystem are nondegenerate and equally spaced, eq 25 can i ck ckk i ∑R dkj (30) be integrated to obtain a closed-form expression for the mean j)1 energy. Assume there are N quantum states with energies 0, were integrated using the fourth-order Runge Kutta algo- - ,2, ...(N 1). The integration of eq 25 gives the simple rithm.31 In our one-dimensional linear chain model, the term result R4‚dkj is replaced by p1dkj/m as discussed earlier. The (N - 1)[1 + /k T - exp(/k T)]k T nonadiabatic couplings dkj between quantum states k and j h) B B B are given by the expression E - (26) 1 exp(/kBT) m Note that the mean energy is proportional to N - 1, the -2 H (xj δ + xj-1 δ ) x p2 k-1,j k-1,j-2 number of quantum states minus 1. Thus, for an infinite ) 2  dkj - number of states, that is, for the harmonic oscillator, the j k equilibrium mean energy of the quantum subsystem is infinite j * k; j ) 1, ...N; k ) 1, ...N at any finite temperature of the classical bath. ) dkk 0 (31)

5. Simulations In the above equation, δk,j is the Kronecker delta. The energy To verify the derived expression for the Ehrenfest mean gap  between adjacent levels was chosen to be 35.9 kJ/ energy of an N-level system with equal energy spacing, eq mol, and mH was 1 amu. Ehrenfest simulations were ) 26, we have carried out numerical simulations. The classical performed for a number of quantum states, N 2, 4, 6, 8, and 10, and for /kBT ranging from 0.54 (high temperature) subsystem is represented by a linear chain of Nc particles, coupled to each other by anharmonic, nearest-neighbor to 2.16 (low temperature). The equilibrium averages of the potentials given by Ehrenfest simulations were obtained from an ensemble of 20 trajectories, each typically 50 ps in length, with a time Nc step e 0.005 fs. The initial 20 ps was neglected in the ) - V(q) ∑VM(qk qk+1) (27) averages to remove any dependence on initial conditions. k)1 The same equilibrium populations were obtained whether where the quantum system started initially in any pure state or in a linear combination of quantum states, confirming that the ) 2 2 - 3 3 + 4 4 system is indeed ergodic and that the averages correspond VM(q) V0(a q a q 0.58a q ) (28) to true equilibrium averages, within statistical uncertainties. > > and qNc+1 is a fixed position. Anharmonic interactions are For low temperatures (/kBT 2) and large N values (N Ehrenfest Mixed Quantum-Classical Dynamics J. Chem. Theory Comput., Vol. 2, No. 2, 2006 233

Figure 1. Mean energy of a quantum oscillator with equally Figure 2. Mean energy of the quantum oscillator with six spaced energy levels, as a function of the number of quantum equally spaced energy levels as a function of inverse tem- states, N. The dashed lines are the mean energies ob- perature. Triangles are the mean energies obtained by the tained by the Ehrenfest method, from eq 26, for /k T ) 0.54 B Ehrenfest simulations. The dotted line is the analytical expres- (---) and /k T ) 2.16 (- - -). The circles and triangles are B sion, eq 26, which agrees with the Ehrenfest simulations within the Ehrenfest mean energies obtained from simulations for statistical uncertainties. The horizontal dashed line is the mean /k T ) 0.54 and 2.16, respectively, confirming the validity of B energy at infinite temperature, i.e., when all six states are eq 26. Note that the Ehrenfest mean energy does not equally populated. The dash-dot shows the classical mean converge with increasing N. The mean Boltzmann energy as energy of a harmonic oscillator, k T. The solid curve shows a function of N is shown for comparison for /k T ) 0.54 (s; B B the energy obtained from a Boltzmann distribution of popula- heavy line) and /k T ) 2.15 (s; light line). B tions for the six-level system. Squares are the results of fewest-switches surface-hopping simulations, as described 6), the length of each trajectory was in the 200-600 ps range, elsewhere,21 showing that surface hopping does achieve the with the initial 100-400 ps neglected, since a longer time correct Boltzmann equilibrium limit for the quantum sub- was required to achieve equilibrium. system.

6. Results Figure 1 shows the mean energy of an N-level quantum system, with equally spaced energy levels, as derived for the Ehrenfest method, eq 26. Also shown are simulation results for N ranging from 2 to 10 and for /kBT ) 0.54 and 2.16. The simulations are in full agreement, within statistical uncertainties, with eq 26. As shown in Figure 1, the Ehrenfest result differs substantially from the desired Boltzmann distribution for an N-state quantum system

N -i/kBT ∑i e Figure 3. Ensemble averaged populations as a function of i)1 h ) time for a quantum oscillator with six equally spaced energy EBOLTZ (32) N levels, from an Ehrenfest simulation. The simulation was - /k T ) ∑ e i B carried out at /kBT 1.44. The quantum subsystem was i)1 started in the ground state (n ) 0). However, the final steady- state populations do not depend on the initial state. Red, The Ehrenfest mean energy does not converge as N increases, green, blue, and magenta correspond to n ) 0, 1, 2, and 3, in contrast to the Boltzmann mean energy, which closely respectively. The horizontal lines show the Boltzmann popula- approaches its asymptotic limit by N ) 10 for the parameters tions at /kBT ) 1.44. Only the first four levels are shown for of Figure 1. clarity.

 exp(-/k T) Figure 2 shows the mean energy of the quantum sub- Eh ) B (33) BOLTZ,∞ - - system, for number of quantum levels N ) 6, as a function 1 exp( /kBT) of the unitless energy /kBT, that is, as a function of inverse The Ehrenfest mean energy also deviates from the classical temperature. As shown in Figure 2, the closed-form expres- expression for the mean energy of a simple harmonic sion for the Ehrenfest mean energy, eq 26, is further verified oscillator, EhCL ) kBT. In the limit, as N f ∞, it is clear by the simulations. The Ehrenfest mean energy is again seen that, at any nonzero temperature, no matter how low, the to deviate substantially from both the quantum and classical mean energy of the Ehrenfest quantum subsystem diverges, Boltzmann mean energies. For comparison, the results of Eh f ∞, as shown in Figure 1. This is a nonphysical fewest-switches surface-hopping simulations, carried out by result. the procedure of a previous reference,21 are also shown in 234 J. Chem. Theory Comput., Vol. 2, No. 2, 2006 Parandekar and Tully

Figure 2. The surface-hopping results agree with the quantum the equilibrium populations of two states to the ratio of Boltzmann results within statistical uncertainty. This offers forward and backward rates. If detailed balance is not additional verification of our previous demonstration that satisfied, then at least one of the rates must be in error, fewest-switches surface hopping satisfies detailed balance possibly affecting even short time dynamics. As demon- rigorously.21 strated previously,21 the alternative, widely used MQCD Figure 3 shows an Ehrenfest simulation of the time method, fewest-switches surface hopping, does not suffer evolution of the quantum state populations for a six-state from this deficiency; at long times, the quantum and classical quantum subsystem of equally spaced levels (only the four subsystems rigorously approach the same temperature. lowest-energy levels are shown), with 50 trajectories in the ensemble. It can be seen from Figure 3 that the Ehrenfest Acknowledgment. This work was supported by the state populations, while not equal, are relatively close in National Science Foundation, Grant CHE0314208. magnitude, in contrast to the Boltzmann populations (hori- zontal dashed lines). References (1) Nikitin, E. E. Theory of Elementary Atomic and Molecular Processes in Gases; Clarendon Press: Oxford, U. K., 1974. 7. Conclusions (2) Jasper, A. W.; Zhu, C.; Nangia, S.; Truhlar, D. G. Faraday We have analyzed the long-time, equilibrium limit of the Discuss. 2004, 127,1. Ehrenfest MQCD method using a Liouville-like equation for (3) Hammes-Schiffer, S.; Tully, J. C. J. Chem. Phys. 1994, 101, the time evolution of the distribution function of the phase 4657. space variables. We find that the Ehrenfest method fails to (4) Staib, A.; Borgis, D.; Hynes, J. T. J. Chem. 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